Probabilistic and Analytical Aspects of the Symmetric and Generalized Kaiser–Bessel Window Function

Theorem 1 The cumulative distribution function of the Kaiser–Bessel distribution has the form Fa,α,ν(x) = ⎧ ⎨ ⎩ 0, x ≤−a (2.5) or (2.6) or (2.7), −a < x < a 1, x ≥a , with the condition that when we use (2.7) we need to suppose that α < 2. Note that since Fa,α,ν(a) = 1, the last case is well-defined, and clearly the

Theorem 2 The effective variance of the Kaiser–Bessel distribution satisfies ℓν + 3π 8 −1 < σν(α) < α 2 2ν+4 ℓν 2(ν + 3) 1 I 2 ν+2(α) + 3π 8 −1 < 3π 8 −1 for all ν > −5 2 and α > 0, where ℓν is described in (3.11).

Theorem 3 If a > 0, α > 0 and ν > −1, then the Kaiser–Bessel distribution has negative excess kurtosis and thus it is a platykurtic or sub-Gaussian distribution. Since the Kaiser–Bessel distribution has compact support, it is not infinitely divisible in the classical sense. The fact that the Kaiser–Bessel distribution is not infinitely divisible in the classical sense can be also deduced immediately from the fact that its excess kurtosis is strictly negative. In other words, if the Kaiser–Bessel d

Theorem 4 The probability density function of the Kaiser–Bessel distribution ϕa,α,ν is increasing on (−a, 0] and decreasing on [0, a) for all a > 0, α > 0 and ν > −1. In other words, the distribution is unimodal, and its mode is zero,7 and similarly as in the case of other symmetric unimodal distributions the mean E[X] = 0 and mode coincide. Moreover, the median of the distribution is also zero because of the symmetry of the support. Figure 1, which resembles to a beautiful colored pashmina, ill

Theorem 5 The probability density function ϕa,α,ν is log-concave on (−a, a) for all for a > 0, α > 0 and ν ≥1 2 and is geometrically concave on [0, a) for a > 0, α > 0 and ν ≥1 2. Moreover, the probability density function ϕa,α,ν is log-concave on (−a, a) for all a > 0, x∗ ν ≥α > 0 and 0 < ν < 1 2, and is geometrically concave on [0, a) for all a > 0, x∗ ν ≥α > 0 and 0 < ν < 1 2. It is worth to mention here that if a probability density function is log-concave, then the corresponding cumulative

Corollary 1 The cumulative distribution function Fa,α,ν is log-concave on (−a, a) for all a > 0, α > 0 and ν ≥1 2, and is geometrically concave on [0, a) for a > 0, α > 0 and ν ≥1 2. In addition, the cumulative distribution function Fa,α,ν is log-concave on (−a, a) for all a > 0, x∗ ν ≥α > 0 and 0 < ν < 1 2, and is geometrically concave on [0, a) for all a > 0, x∗ ν ≥α > 0 and 0 < ν < 1 2. Note that in view of the explicit form of Fa,α,ν(x) a direct proof for the above properties would be not tr

Theorem 6 The following assertions are true: 123

Corollary 2 The next assertions are true: (iii) If a > 0, α > 0, ν ≥1 and a > |x| ≥a · rα,ν−1, then the function x →ϕa,α,ν(x) is convex; if a > 0, α > 0, ν ≥1 and |x| ≤a · rα,ν−1, then the function x →ϕa,α,ν(x) is concave; 10 It is interesting to mention that combining (4.10) with (4.3) we arrive at the following partial differential formula for the probability density function of the Kaiser–Bessel distribution ∂2ϕa,α,ν+1(x) ∂x2 = αx2

Theorem 7 The function ν →ϕa,α,ν(x) is decreasing on (ν0, ∞) for all a > 0, α > 0 and |x| < a, and is decreasing on (ν1, ∞) for all a > 0, α > 0 and a √2ν+1 ≤|x| < a. These monotonicity properties are illustrated on Fig.6. This figure resembles also to a beautiful colored pashmina, and it is interesting to note the similarity between the behavior of the probability density function with respect to α and ν. Figure7illustrates 123

Corollary 3 The function ν →A[ϕa,α,ν] is decreasing on

Theorem 8 The characteristic function of the Kaiser–Bessel distribution is given by φX(t) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ αν+ 1 2 Iν+ 1 2 (α) · Jν+ 1 2 √ a2t2 −α2

Theorem 9 The Kaiser–Bessel random variable X ∼KB(a, α, ν) is determinate via the well-defined moment sequence  μn,ν  n≥1. 5.3 Differential Entropy or Shannon Entropy The differential entropy of the random variable X ∼KB(a, α, ν) is the quantity h[ϕa,α,ν] = E[−ln ϕa,α,ν(X)] = −  a −a ϕa,α,ν(x) ln ϕa,α,ν(x)dx, where ϕa,α,ν is as in (2.2). The change of variable x = a √ 1 −s2 leads to

Theorem 10 The differential entropy of the random variable X ∼KB(a, α, ν) is given by h[ϕa,α,ν] =

Theorem 11 The Rényi entropy of the Kaiser–Bessel distribution is given by R[ϕa,α,ν] = 1 1 −λ ln ⎡ ⎢⎣ a1−λπ 1−λ 2  α 2 λ  ν+ 1 2

Corollary 4 For all x ∈C, ν ∈C\Z−, for the parameters range for which the Kaiser– Bessel distribution KB(a, α, ν) is defined and λ /∈N0 there holds true the summation  k≥1  0<k1<···<kn≤k k1+···+kn=k 1 n j=1 k j!(ν + 1)k j (λν + 1)k (λν + 3 2)k xk =